Not my own ID but attempts to use my address to open credit accounts and illegally purchase goods.
1st attempt last year, courier turned up with a parcel with my address but with a Nigerian name on it. I refused to accept as this is a known scam where the recipient comes by later explaining it's their parcel, they live a few doors away and the dispatchers have made a mistake with the label. Had I accepted the parcel, a month later I would probably have got a huge bill for a mobile phone account taken out at my address.
2nd attempt was successful. I started receiving invoices from a catalog company for my address but in the name of the previous occupant who moved out 15 years ago. No goods have ever been delivered here but the credit account was successfully opened and items ordered/dispatched (where they were sent I don't know). Not wanting my credit rating to get hit I emailed and phoned the company who got their fraud investigators on the case.